Use the graph to determine the values of *a* for which 

lim *x*→*a* *f*(*x*)

 exists. (Enter your answer using interval notation.)

To determine the values of \( a \) for which \( \lim_{x \to a} f(x) \) exists, we need to analyze the continuity and behavior of the piecewise function \( f(x) \) at each segment and at the boundaries where the function definition changes.

### Function Recap:
\[
f(x) = \begin{cases} 
\sqrt[3]{x} & \text{if } x \leq -1 \\
x & \text{if } -1 < x \leq 2 \\
(x - 1)^2 & \text{if } x > 2 
\end{cases}
\]

### Analysis of Continuity:

1. **For \( x \leq -1 \)**:
   - The function \( f(x) = \sqrt[3]{x} \) is continuous for all \( x \) in this interval.
   - Therefore, the limit \( \lim_{x \to a} f(x) \) exists for all \( a \leq -1 \).

2. **For \( -1 < x \leq 2 \)**:
   - The function \( f(x) = x \) is continuous for all \( x \) in this interval.
   - Therefore, the limit \( \lim_{x \to a} f(x) \) exists for all \( a \) in the interval \( (-1, 2] \).

3. **For \( x > 2 \)**:
   - The function \( f(x) = (x - 1)^2 \) is a quadratic function, which is continuous for all \( x > 2 \).
   - Therefore, the limit \( \lim_{x \to a} f(x) \) exists for all \( a > 2 \).

4. **At the boundaries \( x = -1 \) and \( x = 2 \)**:
   - **At \( x = -1 \)**:  
     - From the left, \( \lim_{x \to -1^-} f(x) = \sqrt[3]{-1} = -1 \).
     - From the right, \( \lim_{x \to -1^+} f(x) = -1 \).
     - Since both limits match and equal the function value \( f(-1) = -1 \), the limit exists at \( x = -1 \).
   
   - **At \( x = 2 \)**:
     - From the left, \( \lim_{x \to 2^-} f(x) = 2 \).
     - From the right, \( \lim_{x \to 2^+} f(x) = (2 - 1)^2 = 1 \).
     - Since the left-hand limit does not equal the right-hand limit, the limit \( \lim_{x \to 2} f(x) \) does **not** exist at \( x = 2 \).

### Conclusion:
The limit \( \lim_{x \to a} f(x) \) exists for the following values of \( a \):

- \( a \leq -1 \)
- \( -1 < a < 2 \)
- \( a > 2 \)

This can be written in interval notation as:

\[
(-\infty, 2) \cup (2, \infty)
\]

Would you like to discuss this further or see a visual explanation?

### Follow-up Questions:
1. Why does the limit not exist at \( x = 2 \)?
2. How would the interval change if the function was defined differently at \( x = 2 \)?
3. What is the importance of continuity in determining the existence of limits?
4. How can one determine whether a function is continuous at a point?
5. What is the difference between one-sided limits and two-sided limits?

**Tip:** When analyzing the existence of a limit, always consider the behavior of the function from both sides of the point of interest. If the left-hand limit equals the right-hand limit, the limit exists at that point.

Explore how to determine the values of a for which the limit lim x→a f(x) exists using a detailed analysis of continuity and piecewise function behavior. This problem covers key concepts in limits, continuity, and piecewise functions in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation